Related papers: Lattice-based stochastic models motivate non-linea…
Partial differential equation (PDE) models for infectious diseases, while less common than their ordinary differential equation (ODE) counterparts, have found successful applications for many years. Such models are typically of…
This study delves into the connection between machine learning and lattice field theory by linking generative diffusion models (DMs) with stochastic quantization, from a stochastic differential equation perspective. We show that DMs can be…
Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural…
Diffusion models have recently emerged as powerful stochastic frameworks for high-dimensional inference and generation. However, existing applications to partial differential equations (PDEs) predominantly rely on physics-informed training…
We study a stochastic epidemic model with multiple patches (locations), where individuals in each patch are categorized into three compartments, Susceptible, Infected and Recovered/Removed, and may migrate from one patch to another in any…
Motivated by studies on the recurrent properties of animal and human mobility, we introduce a path-dependent random walk model with long range memory for which not only the mean square displacement (MSD) can be obtained exactly in the…
We introduce and analyze a model for the transport of particles or energy in extended lattice systems. The dynamics of the model acts on a discrete phase space at discrete times but has nonetheless some of the characteristic properties of…
Random walks on lattices with preferential relocation to previously visited sites provide a simple framework for modeling the displacements of animals and humans. When the lattice contains a few impurities or resource sites where the walker…
The exact quantum dynamics of lattice models can be computationally intensive, especially when aiming for large system sizes and extended simulation times necessary to converge transport coefficients. By leveraging finite memory times to…
A space fractional diffusion-like equation is introduced, which embodies the nonlocality in time, represented by the memory kernel and the non-locality in space. A specific example of the nonlocal term is considered in combination with…
Nonlinear mixed effects models represent a powerful tool to simultaneously analyze data from several individuals. In this study a compartmental model of leucine kinetics is examined and extended with a stochastic differential equation to…
Single-particle traces of the diffusive motion of molecules, cells, or animals are by-now routinely measured, similar to stochastic records of stock prices or weather data. Deciphering the stochastic mechanism behind the recorded dynamics…
The nonlocal Fisher equation is a diffusion-reaction equation with a nonlocal quadratic competition, which describes the reaction between distant individuals. This equation arises in evolutionary biological systems, where the arena for the…
L\'{e}vy robotic systems combine superdiffusive random movement with emergent collective behaviour from local communication and alignment in order to find rare targets or track objects. In this article we derive macroscopic fractional PDE…
Progress on modern scientific questions regularly depends on using large-scale datasets to understand complex dynamical systems. An especially challenging case that has grown to prominence with advances in single-cell sequencing…
This paper reports on a data-driven, interaction-aware motion prediction approach for pedestrians in environments cluttered with static obstacles. When navigating in such workspaces shared with humans, robots need accurate motion…
We develop two statistical models for space-time abundance data based on a stochastic underlying continuous individual movement. In contrast to current models for abundance in statistical ecology, our models exploit the explicit connection…
The neural ordinary differential equation (ODE) framework has emerged as a powerful tool for developing accelerated surrogate models of complex physical systems governed by partial differential equations (PDEs). A popular approach for PDE…
Experiments in predator-prey systems show the emergence of long-term cycles. Deterministic model typically fails in capturing these behaviors, which emerge from the microscopic interplay of individual based dynamics and stochastic effects.…
Physical models with uncertain inputs are commonly represented as parametric partial differential equations (PDEs). That is, PDEs with inputs that are expressed as functions of parameters with an associated probability distribution.…